.,2[()] = -[ I ] + X2[1] (1.
2 [1] = X [O] - X2[1. (.
B. CLEAN
The CLEAN algorithm, originally presented by Hbgbom for use in radio astronomy, is an iterative
deconvolution procedure for the removal of unwanted sidelobes [5], [6]. Let the Fourier data of the object
be represented by _(k). The measurements are weighted according to the user defined function _(k). The
image formed by the observations is known as the "dirty map" (DM):
DM = IFFT{(- (k) ()} (1
for a set of K observations. The "dirty beam" (DB) will be defined as the inverse Fourier transform of the
weight function:
DB= IFFT{ (k)}. (1
The maximum points of correlation between the DB and the DM will determine the locations of the real data:
Dl * DB = IFFT{ ~(k) - O(k) - (k)}. (1
For simplification, we will choose to weight the measured points as unity (and all others to zero). Thus,
the correlation function in (17) becomes simply the DM. The CLEAN algorithm is defined as follows. First, the point
of maximum zero deflection is located on the dirty map. A dirty beam pattern, normalized to some loop gain, _,
and centered at this maximum point, is subtracted out from the dirty map. Ideally, the loop gain should be
infinitely small. However, little improvement is noticed for _ less than 0.5, although a longer runtime does result
due to the increased number of iterations [3]. The iterations are repeated until the remaining signal is no
longer significant. Lastly, the removed signals are returned to a clean map through convolution with a clean
beam, typically a Gaussian. The small sidelobes of the clean beam serve to weigh down the higher frequency,
more uncertain terms. When excessive noise is present in the data, CLEAN methods can yield little improvement
in overall sidelobe reduction. In that case, CLEAN would be unable to differentiate between real data and their
noise elements [5].
NUMERICAL EXAMPLES
A 3-D representation of the backhoe tractor can be seen in Figure l(a).